Concentration dependence model

Experimental results presented in this work and in the literature are inconsistent with the assumptions and the physical interpretations implicit in the dual-mode sorption and transport model, and strongly suggest that the sorption and transport in gas-glassy polymer systems should be presented by a concentration-dependent model ... 

The concentration-dependent models attribute the observed pressure dependence of the solubility and diffusion coefficients to the fact that the presence of sorbed gas in a polymer affects the structural and dynamic properties of the polymer, thus affecting the sorption and transport characteristics of the system (3). On the other hand, in the dual-mode model, the pressure-dependent sorption and transport properties arise from a... 

Keywords diffusion through polymeric films, diffusion coefficient temperature and concentration dependence, modelling of diffusion, sorption, crystallinity. 

The usual practice in these appHcations is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be appHed only to problems in which there is a weU-defined, clear association between the independent and dependent variables. The generalization of statistics to the associated confidence intervals for nonlinear coefficients is not well developed. 

Figure 2. The structural energy difference (a) and the magnetic moment (b) as a function of the occupation of the canonical d-band n corresponding to the Fe-Co alloy. The same lines as in Fig. 1 are used for the different structures. In (b) the concentration dependence of the Stoner exchange integral Id used for the spin-polarized canonical d-band model calculations is shown as a thin dashed line with the solid circles. The value of Id for pure Fe and Co, calculated from LSDA and scaled to canonical units, are also shown in (b) as solid squares.

We define a fee lattice and affect at each site n, a spin or an occupation variable <7 which takes the value +1 or —1 depending on whether site n is occupied by a A or B atom. Within the generalized perturbation method , it has been shown that substitutional binary alloys AcBi-c may be described within a Ising model with effective pair interactions with concentration dependence. Thus, the energy of a configuration c = (<7i,<72,- ) among the 2 accessible configurations for one system can be written... 

The commonly used method for the determination of association constants is by conductivity measurements on symmetrical electrolytes at low salt concentrations. The evaluation may advantageously be based on the low-concentration chemical model (lcCM), which is a Hamiltonian model at the McMillan-Mayer level including short-range nonelectrostatic interactions of cations and anions [89]. It is a feature of the lcCM that the association constants do not depend on the physical... 

Although specific calculations for i and g are not made until Sect. 3.5 onwards, the mere postulate of nucleation controlled growth predicts certain qualitative features of behaviour, which we now investigate further. First the effect of the concentration of the polymer in solution is addressed - apparently the theory above fails to predict the observed concentration dependence. Several modifications of the model allow agreement to be reached. There should also be some effect of the crystal size on the observed growth rates because of the factor L in Eq. (3.17). This size dependence is not seen and we discuss the validity of the explanations to account for this defect. Next we look at twin crystals and any implications that their behaviour contain for the applicability of nucleation theories. Finally we briefly discuss the role of fluctuations in the spreading process which, as mentioned above, are neglected by the present treatment. 

Only true rate constants (i.e., those with no unresolved concentration dependences) can properly be treated by the Arrhenius or transition state models. Meaningful values are not obtained if pseudo-order rate constants or the rates themselves are correlated by Eq. (7-1) or Eq. (7-2). This error is found not uncommonly in the literature. The activation parameters from such calculations, A and AS in particular, are meaningless. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

Lyotropic LCs can also be described by a simple model. Such molecules usually possess the amphiphilic nature characteristic of surfactant, consisting of a polar head and one or several aliphatic chains. A representative example is sodium stearate (soap), which forms mesophases in aqueous solutions (Figure 8.4a). In lyotropic mesophases, not only does temperature play an important role, but also the solvent, the number of components in the solution and their concentration. Depending on these factors, different types of micelles can be formed. Three representative types of micelles are presented in Figure 8.4b-d. 

The above model was solved numerically by writing finite difference approximations for each term. The equations were decoupled by writing the reaction terms on the previous time steps where the concentrations are known. Similarly the equations were linearized by writing the diffusivities on the previous time step also. The model was solved numerically using a linear matrix inversion routine, updating the solution matrix between iterations to include the proper concentration dependent diffusivities and reactions. 

The data in Figure 5 can now be considered in light of the conduction model developed above. As stated previously, conduction in reduced poly-I behaves like an activated process. There are two sources that potentially could be responsible for this behavior. The first is the Boltzmann type concentration dependence of the 1+ and 1- states discussed above. The number of charge carriers is expected to decrease approximately exponentially with T. The second is the activation barrier to self-exchange between 1+ and 0 sites and 0 and 1- sites. For low concentration of charge carriers both processes are expected to contribute to the measured resistance. 

On the basis of theoretical calculations Chance et al. [203] have interpreted electrochemical measurements using a scheme similar to that of MacDiarmid et al. [181] and Wnek [169] in which the first oxidation peak seen in cyclic voltammetry (at approx. + 0.2 V vs. SCE) represents the oxidation of the leucoemeraldine (1 A)x form of the polymer to produce an increasing number of quinoid repeat units, with the eventual formation of the (1 A-2S")x/2 polyemeraldine form by the end of the first cyclic voltammetric peak. The second peak (attributed by Kobayashi to degradation of the material) is attributed to the conversion of the (1 A-2S")x/2 form to the pernigraniline form (2A)X and the cathodic peaks to the reverse processes. The first process involves only electron transfer, whereas the second also involves the loss of protons and thus might be expected to show pH dependence (whereas the first should not), and this is apparently the case. Thus the second peak would represent the production of the diprotonated (2S )X form at low pH and the (2A)X form at higher pH with these two forms effectively in equilibrium mediated by the H+ concentration. This model is in conflict with the results of Kobayashi et al. [196] who found pH dependence of the position of the first peak. 

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

Both active and passive transport occur simultaneously, and their quantitative roles differ at different concentration gradients. At low substrate concentrations, active transport plays a major role, whilst above the concentration of saturation passive diffusion is the major transport process. This very simple rule can be studied in an experimental system using cell culture-based models, and the concentration dependency of the transport of a compound as well as asymmetric transport over the membrane are two factors used to evaluate the presence and influence of transporters. Previous data have indicated that the permeability of actively absorbed compounds may be underestimated in the Caco-2 model due to a lack of (or low) expression of some uptake transporters. However, many data which show a lack of influence of transporters are usually derived from experiments... 

The original proposal of the approach, supported by a Monte Carlo simulation study [36], has been further validated with both pre-clinical [38, 39] and clinical studies [40]. It has been shown to be robust and accurate, and is not highly dependent on the models used to fit the data. The method can give poor estimates of absorption or bioavailability in two sets of circumstances (i) when the compound shows nonlinear pharmacokinetics, which may happen when the plasma protein binding is nonlinear, or when the compound has cardiovascular activity that changes blood flow in a concentration-dependent manner or (ii) when the rate of absorption is slow, and hence flip-flop kinetics are observed, i.e., when the apparent terminal half-life is governed by the rate of drug input. 

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